bibkey: hanna2026a396843 authors: Paul D. Hanna year: 2026 title: “OEIS A396843, g.f. satisfying A(xA(x) - 3x*A(x)^2) = x^2” doi: null url: https://oeis.org/A396843 claim: “Conjecture: all terms are odd.” strata_touched:

• D5/S1/Recurrence/Parity/CompositionalSquareAllOddCoefficients license: citation-only triage: anchor

OEIS A396843

Paul D. Hanna’s comment of July 1, 2026 states the conjecture quoted above. The entry defines the ordinary generating function by

A(x*A(x) - 3*x*A(x)^2) = x^2,

with A(x) = Sum_{n>=1} a(n)*x^n. Thus the series has zero constant coefficient. The proof reduces this defining equation modulo two and uses coefficientwise uniqueness together with the explicit reduced solution x/(1+x). It does not use the entry’s other formulas or its separate modulo-nine comment.

Verified locator

• URL: https://oeis.org/A396843